[lit-ideas] [longish, do not read if not interested] Re: fallibility
- From: palma@xxxxxxxx
- To: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- Date: Sat, 1 Dec 2007 12:16:12 -0500 (EST)
Dear Donal McEvoy, thank you for thinking about this.
To be sure, take your time: there is no rush since the primes are
members
of a nonfinite set and they stay that way, and the will --they always did.
The problem I see, quite honestly, with your view is quite serious.
A certain claim, mostly on the basis of popperianism, is being made and it
is, a failure of conjectural capacities imho, extended to a philosophical
thesis.
Noting that, e.g. in your "see the history of Darwinism" remarks, certain
theories have been revised, modified, at times taken to be wrong (nobody,
that I know of, takes seriously the claim that the planet I am on is the
"centre of the universe", whatever that means) the philosophical thesis is
born to the effect that ALL KNWOLEDGE IS FALLIBLE.
I take that to be the thesis put forward (by D. McEvoy,unless I
misunderstand him grossly.)
Now elementary reasoning, as well as popperianism, is bound by the axiom
that a universally quantified statement is falsifiued by (at least) one
instance in which the post-quantifer statement is false. Stop when I say
something unclear.
hence one and even only one case of non fallible knowledge is enough to
make McEvoy's thesis false.
This is fair (by Popper's standards as well as by yours, if in doubt
re-read the discussion about what it takes to falsify the claim that swans
are white: the encounter of one black swan in Australia, as it happens, is
enough. Popper himself made the claim and he lived in Nz, as you may be
aware of.
So I gave you one example of something I know, and it is to the
fallibilist that the burden of proof falls on. For the simple reason that
I do not see what the falsifiability condition is. If you see it, tell me
what it is.
Two points
i/ I chose specifically something that is not trivial (why? that the
naturals are members of omega a nonfinite set is definitionally true,
naturals being recursively defined [for any N, Ns --the successor
function, i.e. +1-- is a member of omega) The primes are not so defined,
hence Euclid, among dozens of others went about finding out whether there
is or there isn't a "largest prime". answer there isn't one.
ii/ you may want to resort to the view that ALL of mathematics is
definitional, but you pay a high price, very high. Namely that we know
once we have PM (Russell & Whitehead magnum opus) we have all of
mathematical knowledge. I won't harass people on why this is false (it is,
in part because of Goedelian reasults and counteless others)
You might also, last ditch -I am playing your role here-- resort to the
view that "advances" in proving techniques will revise that. I believe it
to be tosh. Why? Because we have no revision of any kind in number theory.
What you correctly point out is that new proofs come about and new ways of
doing
the work, as it were, become in fashion within and without the
mathematical communities. The point that, I surmise, escapes you is that
they prove albeit in different ways, the same theorem.
I have chose an elementary one (there are millions of others.)
If you enjoy this kind of problem, do not read pop science by TV
personalities.
Take a look at "Proofs from the book" by M. Aigner & G. Ziegler, Berlin:
Springer: 2004. The chapter on the many proofs of the infinity of primes
is fun and easy.
The only attempt (in "Proofs and refutations") by the late I. Lakatos,
hungarian then british philosopher) to show that there are refutations
is laughable at best.
Note, in case you wonder, that if you found a mistake in the proof(s),
aside form winning many prizes, you have only shown that I did not know
that the primes are members of a non finite set. The condition of
J(ustification) would fail: since I was relying, allegedly, on a faulty
proff, I was not fairly claiming that I knew. In your jargon "Palma
thought that Palma knew that the primes are members of a nonfnite set"
-- and he was wrong in so thinking.
On the philosophical question you ask: do I have any reason to think
that there is such a thing as a humanly-created non fallible proof, I
shall be very brief, we may talk about it in private since the list will
ge annoyed. My own view is that we created NOTHING at all. Like Karl
Popper (and many others) I am a rather rabid mathematical realist.
Humans discover that primes are members of a nonfinite set. The guess,
and this *is* a guess, is that once we discovered it primes do not
change anything or in any way (Popper used to call this the "3rd world")
hence there si nothing to revise. We may, to use an analogy, discover
that we can go to the source of teh Nile wlaking, by Land Rover, or by
chopper, or flying on the wings of eagle, the Nile as it were, does not
change position.
The analogy is faulty because the source of the Nile as much of
geological knowledge is revisable, in view of tectonic plates' shift,
and so forth. This is only a poor analogue of what my guess comes to.
With my best wishes for yoru new year
palma
On
Sat, 1 Dec 2007, Donal McEvoy wrote:
>
> --- palma@xxxxxxxx wrote:
>
> > I would be grateful if you give me the fallibility condition, e.g. under
> > what condition the set of primes would be a finite set.
> >
> > Thank you
>
> This seems a fair question and, while we may hopefully come back to it, I
> hope you do not take this lack of a simple direct answer as showing the
> question is unanswerable [consider: the history of Darwinism] and that the
> point of view questioned is therefore false [consider: the history of
> Darwinism].
>
> First, I am not a philosopher of maths and am of limited maths training (this
> is partly why I was not insulting anyone else's intelligence when I suggested
> - as is often enough written - that Newton's theories are, as written in
> 'PM', beyond most people because expressed at a certain level of mathematical
> symbolism that is beyond most people).
>
> Second, the question arguably contains a 'trap'. In effect it puts the burden
> on the fallibilist to say what would be the "fallibility condition" for a
> certain theory or theorem. But is this fair? The fact is, in both the history
> of empirical theories and mathematical theorems, until the mistake has been
> spotted we are not necessarily in a position to state in advance what the
> mistake might or will be. That is, even if all theories are fallible it may
> be impossible to give in advance a "fallibility condition" for a specific
> theory or theorem. If you doubt this, take any empirical theory or math
> theorem that has been overturned by some subsequent "advance" and explain how
> - in advance of that "advance" - you or anyone else might have stated the
> "fallibility condition"? Not easy, yeh?
>
> Third, problems of infinity in maths are notoriously difficult. I may be
> wrong but is the problem of an infinity of primes really that distinct from
> the problem whether there is an infinity of natural numbers: we know primes
> get (generally) further apart the further we go along the list of natural
> numbers, but if that list is infinite why shouldn't the string of primes be
> infinite? And, yes, how would you set out to prove or disprove such a thing?
> And would that mean there was such a thing as a human-created "infallible"
> proof or disproof?
>
> I've read Singh's "Fermat's Last Theorem" and it is [afair] sketchy as to the
> actual "proof". Indeed the writer of the "proof" believed he had earlier hit
> on a "proof", and believed this without having a "fallibility condition" for
> that earlier "proof". Nevertheless, he knew he might be mistaken - _and with
> this critical attitude towards his own work_ saw eventually that his first
> "proof" was inadequate.
>
>
> Donal
> Treading in deep waters
> Fearful of sharks and thingies
> And of doing a tummy-shame
>
>
>
> __________________________________________________________
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>
>
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